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Platzhalter für Bild, Bild auf Titelfolie hinter das Logo einsetzen
Dr. Noemi Friedman, 31.10.2014.
Introduction to PDEs and Numerical Methods Tutorial 2: Analytical solution to PDE’s
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 2
Overview of this tutorial
Introduction: Differential operators Classification of PDEs Examples of ODEs/PDEs Eigenvalues, eigenvectors
Analytical solution of ODEs (1D – Poisson equation) Some important notations Heat equation Analytical solution to the Poisson equation by integration Solution with the Green’s function – the fundamental solution
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 3
Some important notations Important vector spaces • Euclidian n-space:
• C[a, b] : set of all continuous, real-valued functions defined on the interval [a, b]. • C1 [a, b]: set of all real-valued, continuously differentiable functions defined on
the interval [a, b]. (A function is continuously differentiable if its derivative exists and is continuous.)
• Ck [a, b]: space of real-valued functions defined on [a, b] that have k continuous derivatives
Dirichlet
Subspaces
Neumann
M.S. Gockenbach: Partial Differential Equations: Analytical and Numerical Methods, Chapter 3: Essential linear algebra!!!
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 4
Heat equation 1D
Derivation of heat equation: Lecture Script, Chapter 1.1
homogeneous (no external heat source)
rhs/source term
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 5
Heat equation 1D
Derivation of heat equation: Lecture Script, Chapter 1.1
homogeneous (no external heat source)
rhs/source term
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 6
Heat equation Poisson equation
tenders towards an equilibrium state:
(Laplace equation)
𝑓 𝑓 (Poisson equation)
1D Poisson equation:
Dirichlet B.C.
Neumann B.C.
𝑝(𝑥)
𝑙
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 7
Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)
𝑙
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 8
Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)
𝑙
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 9
Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)
𝑙
𝑥3
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 10
Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)
𝑙
𝑥3
31. 10. 2014. | Dr. Noemi Friedman | PDE tutorial | Seite 11
Analytical solution of the Poisson equation 1D Poisson equation: 𝑝(𝑥)
𝑙
𝑥3